Thanks Jay! I have wondered why I got a bit stuck on the whole URL / DNS / LetsEncrypt thing in the (recent) past when attempting to follow the instructions in the OP.
I got there in the end by using a subdomain for the new site, ensuring my restored site in the new server was working, and then doing a quick DNS / URLs switch. But that was all a bit sketchy / painful (especially the remap whack-a-mole afterwards!).
Why I needed to do all that makes sense now. And good to know that rsync can sidestep some of the pain points.
I think that some clear instructions here would really help others, as it is a bit confusing at the moment - and I’m not sure the best way to do it in future. Could your disclaimer perhaps be rolled into the whole OP as a re-write (perhaps by @SaraDev?).